Three-Dimensional Virial Theorem (Quantum Mechanics)

[NeoDevin]

Homework Statement

(a) Prove the three-dimensional virial theorem:

$$2<t> = <r\cdot \nabla V>$$

(for stationary states)

Homework Equations

Eq. 3.71 (not sure if this applies to 3 dimensions, but I think so)

\frac{d}{dt}<Q> = \frac{i}{\hbar}<[\hat H, \hat Q]> + \left<\frac{\partial \hat Q}{\partial t}\left> [/tex]

where the last term is the explicit time dependence of the operator Q.

The Attempt at a Solution

Letting $Q = \vec r \cdot \vec p$

$$\frac{\partial \hat Q}{\partial t} = 0$$

and for stationary states:

$$\frac{d}{dt}<Q> = 0$$

so:

$$0 = \frac{i}{\hbar}<[\hat H, \hat Q]> = \frac{i}{\hbar}<[T+V, \vec r \cdot \vec p]>$$

$$= \frac{i}{\hbar}(<T(\vec r \cdot \vec p)>-<\vec r \cdot \vec p T> + <V(\vec r \cdot \vec p)> - <\vec r \cdot \vec p V>)$$

but

$$<\vec r \cdot \vec p V> = <\vec r \cdot (\vec pV)> + <V(\vec r \cdot \vec p)>$$

so

$$0 = \frac{i}{\hbar}(<T(\vec r \cdot \vec p)> - <\vec r \cdot \vec p T> - <\vec r \cdot (\vec p V)>)$$

 

[kuruman]

First you show that
$$\begin{align}\frac{d\langle \mathbf{x}\cdot\mathbf{p}\rangle}{dt}=-\frac{i}{\hbar}\sum_{j=1}^3\left(\langle x_j[p_j,H]+[x_j,H]p_j\rangle \right).\end{align}$$
Second you show that $$\begin{align}[p_j,H]=+\frac{\hbar}{i}\frac{\partial V}{\partial x_j}~~\text{and}~~[x_j,H]=-\frac{p_j}{m}.\end{align}$$
Third you substitute equations (2) into (1) and it should pop out.